The arbitrary position control of cylinder has always been the hard challenge in pneumatic system. We try to develop a cylinder position servo control method by combining fuzzy PID with the theoretical model of the proportional valvecontrolled cylinder system. The pressure differential equation of cylinder, pressureflow equation of proportional valve, and moment equilibrium equation of cylinder are established. And the mathematical models of the cylinder driving system are linearized. Then fuzzy PID control algorithm is designed for the cylinder position control, including the detail analysis of fuzzy variables and domain, fuzzy logic rules, and defuzzification. The stability of the proposed fuzzy PID controller is theoretically proved according to the small gain theorem. Experiments for targets position of 250 mm, 300 mm, and 350 mm were done and the results showed that the absolute error of the position control is less than 0.25 mm. And comparative experiment between fuzzy PID and classical PID verified the advantage of the proposed algorithm.
In 1956, Shearer [
With the development of computer technology and modern control technique, the pneumatic servo control problem was revisited by scholars. Scavanda et al. [
In this paper, we investigated a proportional valvecontrolled cylinder system and developed a position control method. Firstly, nonlinear mathematic model of the cylinder is established in Section
The dynamic characteristics of cylinder are mainly described by three equations: the pressure differential equation of cylinder, pressureflow equation of proportional valve and moment equilibrium equation of cylinder.
The flowing state of air inside the pneumatic system is extremely complicated. To simplify the system mathematical model, we use the following hypothesis.
The working media (here refers to air) in the system is taken as ideal gas.
The flowing state while the air runs through the valve port or other chokes is taken as the isentropic and adiabatic process.
The lumped parameter model is adopted, ignoring the influences on the system from the distributed resistance in the air tube and flexibility of the pipeline.
The air pressure and temperature inside the same chamber are equal everywhere.
There is no leakage of the cylinder, both inside and outside.
The pressures of air source and atmosphere are constant.
We suppose that the flowing air inside the thermodynamic system has no energy exchange with the outside and the pressure changes slightly, during the fast inflating process from air source to cylinder chamber. And then, this flowing process can be taken as the isentropic and adiabatic process. According to the energy equation of adiabatic inflating process from constant pressure air source to limited volume, there are four kinds of energy changing processes inside the volume during the movement [
The air will bring in or take out the energy
The flowing work between the volume and the outside during the air runs in and out of the chamber is
The thermoexchange between the chamber and the outside is
The work from the chamber to the outside during the piston movement is
If we ignore the leakage of cylinder and valve, according to the energy conservation principle, the total internal energy
Supposing that the gas is ideal air and disregarding the kinetic energy and static energy of the air, we can get
As is well known, the internal energy of air is
Substituting the above equations by formula (
Generally, the rate of heat exchange
In the proportional valvecontrolled cylinder system, the air mass flow running into and out of the cylinder chamber is controlled by the port area of the proportional valve. And the air mass flow
We can obtain the kinetic equilibrium between the cylinder and load by the force analysis for the system
Combining the Coulomb friction and external load as
From the above dynamic characteristics basical equations, it is clear that the system is nonlinear. So we linearize the system near the cylinder equilibrium point based on the linear system theory.
Generally, the spool opening area of proportional servo valve can be taken as the linear function of the controlling voltage; that is, the spool displacement is directly proportional to the controlling signal:
Linearizing the flow equation of the proportional valve and applying the Laplace transform, we can get
Linearizing the pressure differential equations of the cylinder chambers (
The force equilibrium equation (
From the above analysis, the cylinder position servo control diagram can be drawn as Figure
Cylinder position servo control diagram.
If
Substituting the above equations into (
PID algorithm is the most used and useful control technique in mechatronics system. But the classical PID algorithm has its inherent shortcomings in practice because of the fixed parameters. For example, the fixed parameters cannot take into account the dynamic features and control requirements in both transient process and stable period. It often fails to achieve the ideal integrated control quality. So, in practice, PID algorithm is usually combined with other parameter adjusting methods, such as fuzzy logic and artificial neuro network.
We integrate the classical PID algorithm and fuzzy logic, using fuzzy logic to adjust the PID control parameters according to the deviation and its gradient between the output and target. Thus we can control the cylinder position precisely. The basical control principle is shown in Figure
Fuzzy PID control principle.
The PID control input is
The deviation
The triangle membership function is adopted, and the membership function for
Fuzzy logic rule for

 

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Using the above fuzzy logic rules, the PID control parameters can be adjusted as
Define
Then the fuzzy relations of
The outputs of the fuzzy logic rules are also fuzzy set. In practical digital control system, the parameters must be defuzzified, that is, converting the fuzzy set into exact values according to an appropriate algorithm.
We use conventional gravity center method to realize the defuzzification:
It is obvious that the calculating process needs certain time, which makes it difficult to be used in realtime control system. So, the calculating process is executed offline in advance. Then the produced defuzzification decision tables are stored in the memory of the controller. In this way, the instantaneity of the control process can be enhanced.
Chen and Ying [
As described in Section
Discretetime Fuzzy PID controller.
The stability of the fuzzy PI controller and the fuzzy PD controller has been analyzed in [
A sufficient condition for the nonlinear fuzzy PI control system to be globally boundedinput and boundedoutput (BIBO) stable is that
the given nonlinear system has a bounded norm (gain)
the parameters of the fuzzy PI controller
where
In the same way, by disconnecting the fuzzy PI controller from Figure
A sufficient condition for the fuzzy
Till now, we are sure that the fuzzy PI controller and fuzzy
Again, the Fuzzy PID controller shown in Figure
Equivalent closedloop control system for the fuzzy PID controller.
Another form of the fuzzy PID controller
Equivalent structure of the controller
The experimental system is composed of pneumatic servo control actuating mechanism, feedback units, loading module, and controller. The pneumatic servo control actuating mechanism is symmetrical cylinder system controlled by proportional flow valve. The feedback units include displacement transducer and the pressure transducer for the cylinder chambers. The whole controller for the system includes industrial personal computer (shorted as IPC), A/D, and D/A board cards for data acquisition and output. The experimental system schematic diagram is shown in Figure
Experimental instruments.
Name  Model  Specification  Brand 

Main cylinder  CA2WL40500 

SMC 
Flow proportional valve  MPYE51/8010B  Max flow: 700 L/min, response: 3 ms, lag: 0.3%  Festo 
Pressure proportional valve  MPPE51/8010B  Max flow: 820 L/min, response: 3 ms, lag: 0.3%  Festo 
Displacement transducer  MTS500  Range: 500 mm, resolution: 5 us, repeatability: ±0.001% FS  MTS 
Pressure transducer  JYBKOHVG  Accuracy: 0.25% FS, range: 01 Mpa, response: 30 ms, nonlinearity: ±0.2% FS, repeatability: ±0.1% FS  Kunlun Coast 
Force transducer  BK1  Range: 1500 N, accuracy: 0.05% FS, nonlinearity: 0.05% FS, repeatability: 0.05% FS  Kunlun Coast 
Pneumatic servo control system principle.
Experimental system.
The control software was developed based on MATLAB and LabVIEW. All the fuzzy logic and PID control algorithms were realized in MATLAB simulink toolbox and then compiled into realtime control program using RTW technique.
RTW is an important supplementary functional module for MATLAB graphic modeling and simulation module Simulink. Optimized, portable, and personalized codes can be directly generated from Simulink model with RTW tools. According to the specific target preparation, the generated codes can be compiled into program for a different rapid prototype realtime environment. RTW ensures us to focus on the model establishment and system design and release from the boring programming work. This kind of developing pattern is very suitable for laboratory experimental system design.
RTW technique has the following features: (1) it supports continuous, discrete, and hybrid time system, including conditioned executing system and nonvirtual system; (2) RTW seamlessly integrates the RunTime Monitor with the realtime target, which provides an excellent signal monitor and parameters adjusting interface. The flow diagram of realtime control program developing using RTW technique is shown in Figure
Working flow with RTW.
LabWindows/CVI is adopted to create the control program frame and user interface, shown in Figure
LabVIEW control program diagram.
On the experimental platform, we set the target position of the cylinder as 250 mm, 300 mm, and 350 mm, respectively. And the control results are shown in Figures
Response of target position 250 mm.
Response of target position 300 mm.
Response of target position 350 mm.
The rising times of the three experiments are 2.65 s, 4.3 s, and 3.2 s, respectively, which indicates that long displacement does not mean long corresponding time. During the motion, the proposed fuzzy PID controller can adjust the control parameters and change the behavior of the system to achieve the best performance. Also, the overshoots in Figures
Control errors of cylinder position (mm).
Initial  Target  AE  RE 

100  250  0.2441  0.20% 
100  300  0.20  0.07% 
100  350  0.2441  0.09% 
AE represents absolute error and RE denotes relative error.
From the experimental data, three significant features can be drawn as follows.
Dynamic quality: the proposed method has fuzzy logic virtues in the earlier stage of control that can actuate the cylinder to approximate the target position rapidly. And during the late stages of control, it has virtues of PID algorithm, which means that the PID parameters are adjusted to execute the cylinder to quickly reach the target position without overshoot.
Stable quality: the analysis of stable error is listed in Table
No creeping phenomenon: when the cylinder runs with quite low speed or stops in the middle, there will be creeping phenomenon because of the air pressure in both the chambers and friction. From the response data in Figures
To show the advantages of the proposed cylinder position servo control method, an experiment was done to compare the classical PID controller and the developed one in this paper, with the target position 300 mm.
The stable state data and error data are shown in Figures
Stable data of comparing experiment.
Error of comparing experiment.
The nonlinear mathematical models of cylinder and its valvecontrol pneumatic system, that is, pressure differential equation, pressureflow equation, and moment equilibrium equation, are proposed.
The cylinder position servo controller based on the mathematical models and fuzzy PID algorithm is established and proved to be stable under specified conditions.
Experimental results show that the absolute control error is less than 0.25 mm and the proposed fuzzy PID controller has better performance than classical PID. The dynamic and stable qualities of the controller are quite well.
The authors wish to confirm that there is no known conflict of interests associated with this paper and there is no conflict of interests for any of the authors.
This work is financially supported by the National Natural Science Fund of China (Grant no. 51275470), the open fund of Key Laboratory of E&M (Zhejiang University of Technology), Ministry of Education & Zhejiang Province (Grant no. 2011EM001), and fund of Zhejiang Educational Department (Grant no. Y201225592).